1

U.S. Particle Accelerator School

Fundamentals of Detector Physics

and Measurements Lab - IV

Carl Bromberg

Michigan State University

Dan Green

Fermilab

June 18-22, 2012

Outline

Lecture I

Constants, atoms, cross sections

Photoelectric, TOF

PMT, SiPM Scint, Cerenkov

Lecture II

Collisions, cross sections

Multiple scattering, radiation length

dE/dx, MIP, Range

Critical Energy

2

Outline II

Lecture III

B fields, trajectories

Quadrupoles, focal length

Drift and Diffusion

Pulse formation in unity gain and gas gain

Lecture IV

Radiation NR, Thompson, Compton

Relativistic radiation

Bremm, Pair Production

3

Si Vertex and b decays

4

~ / 2

~ / 2

a

T D

a

D

D a b

p M

p p

~ / ~1/

~ ( )

D D D

D

M p

b c

22

2

/22

/2

2

, 0

/

/

/12

P

P

y y y

y dy dy

y dy dy

P

HEP collider detectors have

“vertex detectors” which serve to

tag the production and decay of

heavy flavors. The “impact

parameter” b, is ~ the proper

decay length of the parent. The

spatial resolution should be << b.

b Tagging

5

US PAS, June 18-22, 2012

Impact parameter, decay length and mass of tracks at the

decay point are all possible tagging variables.

CMS 7 TeV – Heavy Flavor, (b)

Note error ellipses on the primary and secondary vertices.

Typical operating point is 60% efficient with a light flavor

rejection of 100.

6 US PAS, June 18-22, 2012

Intrinsic/Doped Si

7

12322 10~/,/1001.5~ siio

si nncmA

Nn

i

ii

KTii

cmknq

cmpn o

/1

200/1

/10300

311

~ 3.6

~ 300

~ /

~ 5 ~ 32,000

pair

s ion pair

s

E eV

d m

q q E E

q fC q

Intrinsic charge is due to thermal

excitation of a e-h pair. Since kT

~ 1/30 eV and Eg~ 3.6 eV this

charge is very small and intrinsic

Si has a high resistivity.

Therefore very low levels of

donor(n) and acceptor (p) states

can determine the resistivity and

the majority charge carriers

A typical detector has a MIP

signal of ~ 5 fC or ~ 32 thousand

e-h pairs in the VB-CB.

Si Energy Bands

8

CB (MT) n (e-)

VB(full) p(h)

-D

- A

Eg

Excite e- to CB from donor states

in gap, leave fixed + ions

Excite e- into acceptor states

leaving mobile holes in VB. The

– fixed states remain

- + - ---------

------------ +

+

n p

Incident

particle

Si Diode

9

/

1

300 ,1

( ) 1

/ 25 o

qV kT

o

F K mA

I V I e

kTR dI dV

qI

“Dope” Si with a small

fraction of donor/acceptor

sites which are still >>

intrinsic conductivity. Form a

diode from a sandwich of n

and p type.

Io = reverse

current

(minority

carriers). V< 0

Forward bias, V

> 0 (majority

carriers)

Si Diode - II

10

DnAn DA Charge is conserved

[ ] / , ( )

D

E qn C C or CGS

CDxD

Anq

CDxnqxE

CAxnqxE

A

D

A

/)(

/)()0(

/)()0(

Applied voltage => depletion region.

Charge swept to electrodes. Static

charge remains, p type < 0, n type >

0.

Si Diode Fields - Depletion

11

2

2

( )( 0)

2

( )( ( 0) )

2

A

AD

E V

q n A x DV x

DC

q n x AV x V

C

2 22/ 2 , ,DV qp d C qp d MKS

potentials

2~ / 2D AV qn d C

d~A, fully depleted when

all free carriers are swept

to electrodes.

Derive V from E.

Just at depletion the

electric field is 0 at

the electrode

location

Si – Pulse Formation

12

2

/

2( )

( ) 1 , / ,

(0) 0, ( )

D

D

t

D

Vdx Edt x d dt

d

x t d e C qp

x x d

2

2 / 2 /

2

( ) / / ,

42 /D D

s D

t ts Ds D

I t dQ dt q E V

q Ve q e

d

2

( ) 2 /

( ) 0

/ 2

/ (0)

s D

D D

I o q

I

d V

d E

)240(710/~)(

sec)20(sec7~

5

nAnAqoI

nn

fCq

Ds

D

s

Si is a unity gain device, so the treatment is as before.

Just at depletion

E field is 0 at x=d, drift

velocity

Line ionization over

distance d, t = arrival

time at electrode

Time to collect charge at

depletion = distance/drift

velocity @ x=0.

e, h

Si Pixels

13

Present pixel generation = diode pixels bump bonded to readout chip.

3-d Silicon

14

Basic cross section ~

1/M^2 so a factor 2x

in mass reach is at

least a factor 10 x in

luminosity – and PDF

falloff makes it often

very much worse.

Upgrade @ LHC will

be a large increase in

luminosity. The

geometry of vertex

detectors will evolve.

One option is a 3-d

detector.

Vertex Detectors

15

( ) 87

( ) ~ 475

( ) ~ (123,312) ( , )

b

o

c

c m

c m

c m D D

Pixel size scale set by

the lifetimes. PU is a

problem so that means

occupation of a pixel

must be small in order

to do robust tracking.

H->Z+Z->e+e+u+u

Tracking - Pixels and Strips

16

“vertex” pixels ~ 200 um x 200 um.

Silicon strips ~ 200 um x 20 cm.

For V=50 V, d = 300 um, uE = 42

um/nsec, time~ 7 (21) nsec for e

(h). 100 M pixels, |y| < 2.5.

US PAS, June 18-22, 2012

Readout -Noise – Reverse I

17

The reverse (minority carrier) diode leakage current is a strong

(exponential) function of T, exp(qV/kT). The scale is ~ nA.

Radiation damage makes defects => impurity states in the diode gap

=> enhanced leakage current. Run detectors at low T.

High resistivity, back biased Si

is ~ a current source.

Noise – Thermal and Shot

18

d

R

kTdI

T

22

dqIdI s

2

2( ) [ /1 ( ) ]f G

Must limit the bandwidth to limit the

noise - filter

Thermal energy kT=> thermal

power. Shot noise due to

quantized q of I => fluctuations

Base Resistance Noise V

19

/ 1/ ,B E mR kT qI g

)(09.0

2

,)(4.0

R

HznA

R

kT

HzAInAqI

d

g

kT

C

qI

CR

kT

RIdZIdZIdVd

mS

B

SS

BTCSSCST

2

)(

222

2222222

2 2 2

2 2

( ) ( / )

/2 4 2

BS

S m

V f dV d d

qIkT kTG C

R g

Base current of front end

transistor thermal noise

rms voltage due to

thermal source resistance,

base current shot noise

and thermal base

resistance.

Output voltage after the

bandwidth limiting filter.

Numerical values for shot

and thermal noise

ENC – Series and Parallel

20

2 2( / )sENC G C e V

2

2

/ / 42

/ / 2

S P B

S S s m

kTENCP C e G V e q I

Rs

ENCS C e G V eC kT g

An input charge qs becomes a voltage at the output at

the frequency peak of the filter, in “electron units”.

Compare the signal to the Equivalent Noise Charge (ENC) referred to

the input.

~ ( / ) /s SV q G C e

~1/

The “parallel” noise

depends on the shaping

time and is due to thermal

source resistance and base

current shot noise. The

“series” noise depends on

inverse shaping time,

source capacity and thermal

base resistance noise.

signal

ENC – Parallel, Series and Shaping

21

Optimize the shaping time for a

particular application. The 2

contributions have different shaping

time dependence.

ENC vs Input Capacity

22

Best Operation

•Low T

•Large Rs (current source)

•Small base current

•Small source capacity

•Small base resistance

•Short shaping time (HEP)

•Good front end transistor

ENC vs. Shaping Time and Total ENC

23

2 2/ / ~ 23

( ) / ~ 800

s

n s

S N q ENCS ENCP

Front end noise e C e

( ) ~ / , ~1 /n s nENCS e C e nV Hz

Example:

Source charge = 5.1 fC

1

1

300

(25 ) , 30 , 10 sec

1125

1 ,,

825

o

m S

S

S B mA

P

kT K

g C pF n

ENC e

R M I

ENC e

Noise

from

front end

transistor

Demo – Radioactive Decay

24

10 year lifetime

Data fitting – with constant

errors, using a MATLAB

utility.

NR Dipole Radiation

25

2 2

22 2

2 2

2

3

2 3

sin~

~ ~ sin / ,

~

sin1/

4

2/

3

q arE

r c

u E qa rc energy density

P cr E

qad P d

c

P qa c

Translate electrostatic

results to NR radiative

solutions using the

addition of a factor =

dimensionless quantity –

acceleration causes

radiation.

Power through a

sphere is

independent of r =>

radiative solution.

Power is ~ square of

acceleration.

Dipole Power

26

, 2 [ sin ]b d dipole moment D q b d t

2sin ~1/ 3(8 )d

2 2 4 3

24 3

23 2

/

/ 3 ,

/ 3 , /

P q d c

P D c

D c D D t

Relate the acceleration to

the size of the radiating

system and the frequency.

Result is that power * c^3

is = dipole moment

acceleration squared

Thompson Scattering

27

2

, ( )4

8o

cS E B CGS

cS E

22 2

3sin

4

/o

d P ea

d c

a eE m

22 2 2

2

/ sin / /

8

3

T

T

de mc d P d S

d

2 4

0

2 4 8

0

8

3

/ ( )

/ ~ ~ 10

T

o

T

a

a Section I

a

Poynting vector

Dipole radiation

Scattering viewed as

the radiation of

incoming power

Cross section is Compton

wavelength squared times

fine structure constant

squared. Much smaller the

atomic cross section

Thompson Form Factor

28

01)(

)(2

ur

rUkdr

ud

RY~ function, waveradialR Rr,u

2

2

2

m

1u rr o

2

o 2

2

4~ sin

k

~4 a

Scattering off an object

with internal structure.

Wave function near r=0 is

defined by centrifugal

potential

Phase shift goes as a

power of ka, if ka < 1 S

wave dominates

Structure has a

characteristic size ~ a.

Black sphere in this case,

totally absorb r<a.

2 1~ ( )

~o

ka

ka

Form Factor - II

29

Why is the sky blue ?

Scattering of the blue while

transmitting the yellow

wavelengths. Rayleigh

scattering.

Wave number

For square

well

Phase shift, S

wave

Small Uo

approximation.

“Form factor:

due to

structure of

scattering

objects.

Compton Scattering

30

UR scattering of e - e

SR Thompson/Compton

Scattering

In Thompson scattering the

radiation emitted has the

same frequency as the

incident wavelength.

However, as the photon

wavelength -> the e

Compton wavelength, the

emitted photons have a

lower energy as the recoil e

take off substantial energy.

The cross section then fall

with energy, in a fashion

clear from the Feynman

diagram.

Compton – II, Kinematics

31

)cos1(2

cos1111

o

o m

SR conservation of

momentum and energy.

Energy outgoing is <

incident energy. The

shift in wavelength is ~

the Compton wavelength

and angle dependent (no

forward shift)

Compton and QM

32

2

2* ~ / 2 ~ / 2, *( * / ) ~ *

2

oo

mp s m d d p p d

The dynamics (t

channel exchange)

leads to large energy

transfer to the e.

Transform from CM to lab frames. SR “searchlight throws everything

forward.

Demo – Angular Distribution

33

Low energy is ~

forward/backward

symmetric. At

high incident

energy, the final

state particles are

thrown forward.

Special Relativity

34

( , )x x ct

cv

csddxU

,

)/(/

2 2 2 2

2

( ) ( / )

1/ 1 , /

ds dx dx dx cdt cdt

v c

2

( , / )

( , ), /

/

p mU p c

m v c v dx dt

mc

p mc c

Formulated as 4

dimensional tensorial

objects – position,

velocity, momentum.

Instead of classical

mechanics and time as a

label, now need a

relativistic invariant –

proper time ds.

Momentum and energy

are conserved – not

kinematic energy due to

the rest energy.

35

Demonstration – Time Dilation

Tick/Tock – in

rest frame time

is measured on

a single clock

– proper time.

In the moving

frame the time

is measured on

spatially

separated,

synchronized,

clocks.

Basically, the

length the light

travels is just

longer.

Acceleration in SR

36

3

3 2 2

/ ( / ),

[ ( , )]

( ), /

[ ( ), ( )]

A dU d s c

dv c

dt

da a dv dt

dt

A a a a

Acceleration in SR is a

complex object when

formulated in terms of

local clocks and rulers.

In SR acceleration is the

proper time rate of change

of the 4-velocity.

The SR power radiated by

a charge due to

acceleration is the

“length” of the 4-

acceleration.

Radiation in Linear/Circular

Acceleration

37

243

2

3

2a

c

eP

2

32

2

3

2

,1,

dx

d

cm

eP

cdtdxdx

d

mA

LIN

6232 )/( 3/2 aceP LIN

The power radiated in

circular motion is much

larger than expected in

NR case. Note – that is

why e accelerators are

linear recently.

In the linear case, the

required power is simply

proportional to the lab

energy supplied per unit

length – e.g. r.f. cavities.

Searchlight Effect

38

)( * *cos *) *(1 cos *p

2

2

2

2

11

1

1 2

12

2(1 )

2

2

2

1 cos ~ 1 (1 / 2)

1 1~

2

~ 1/

Transform the energy of the

radiation from the C.M>

(starred) frame to the lab

observer frame.

For a more isotropic

distribution in the C.M, the

radiation in the lab frame is

typically localized in a

small angle region. This is

called the searchlight effect.

Demo - Radiation

39

linear circular

2 3 2 5

2 3 2

/ / 4 sin / (1 cos )

/ / 4 (sin )

LIN

LIN

d P d a c

d P d a c

NR dipole radiation

goes forward in

both the linear and

circular case.

Demo – “Searchlight”

40

The sharp rise

of the total

radiated power

with energy is

very evident,

as is the

forward nature

of the angular

distribution.

Circular Acceleration

41

3

3

~ /

~

3 / 2

c

c a

c a

3 4

2

4~ ~~

3

~

~ ( ) /

o

c

aP

c

c

dI d

The NR frequency is given by the velocity and the radius. The SR

frequency (time dilation) rises rapidly with energy and extends up

to a critical frequency. The energy loss due to radiation is the power

times the lab time interval. This leads to the fourth power. The

frequency spectrum is ~ flat out to the critical frequency.

Synchrotron Radiation

42

An e in a magnetic

field spirals around

the field line while

emitted synchrotron

radiation => smaller

radius of curvature –

the “death spiral”.

Numerical values for

e at bend radius a.

4

3

c

( ) 90[ ( )] / ( )

( ) 2.2 / ( )

E keV E GeV a m

keV E GeV a m

Demo – Circular, SR

43

As before,

the dipole

like pattern

become

much larger

in absolute

power

radiated and

become

much more

forward

peaked.

Pair Production

44

An isolated charged

particle cannot

radiate (good idea

or we would not

exist). A photon can

virtually decay into

an electron-positron

pair by using energy

from the photons in

the Coulomb field

of a nucleus. As the Compton scattering cross

section falls off with photon

energy, the pair production

becomes the dominant energy

loss mechanism for photons at

high energies.

Pair Production - II

45

7`

9pair B

22~ ln( )eB Z

3/ / (1/ )~

/ (1/ )

/

pair

B

d dy y

d dy y

y E E

Pair production and Bremmstrahlung are closely related processes

(later). They are coherent over the small size of the nucleus and go

like the third power of the fine structure constant – see Feynman

diagram. Pair production favors soft electrons.

Note the “death

spiral’

Bremm – WW Virtual Quanta

46

2

2

/

dU du E d bdb

dU d d

2/ 4 / lndU d

dU dN

2

2

( ) 1~ [ln( )]

2 1 1[ln( )]

dN

d

The nucleus can be viewed

as a source of virtual photons

which then scatter incoming

charged particles. Recall that

a Coulomb collision has

energy loss ~ the inverse of

the velocity squared.

The energy can be converted

from energy and frequency to

the number of virtual photons

of a given frequency. The

basic dependence of the

spectrum is that it falls as the

inverse of the photon energy.

Bremm - II

47

2

2

2

~

~ ln ( )

BT

Be

dNdZ

d d

Zd

d

~ ( )E

BN dx d

dE dA d

The Feynman diagram is closely

related to that for photon pair

production. The nuclear

Coulomb field supplies the

virtual photon which allows the

electron to radiate a photon and

still conserve energy/momentum.

The Bremm frequency spectrum

can be (NR) viewed as the

Thompson scattering of the soft,

virtual photons of the nucleus. The

energy loss of the electrons has a

characteristic inverse photon

energy emission spectrum.

coherent

Radiation Length

48

/1 1, ( ) ( )

x XdEE x E e

E dx X

1 2 216( )( ) [ln( )]

3

~ ( / ) ~

e

NX Z

A

Z A Z Z

[ / ( )]B

A N X

X L

Xo(gm/cm2) = [180(A/Z)]/Z.

In Lecture I the radiation length appeared – arbitrarily. Here it

is defined to be the mean free path to emit a photon. It goes as

Z (coherence less 1/A) times the third power of alpha

(counting vertices).

Muon Energy Loss

49

Muons are just heavy electrons (so far), responding to the same forces

as electrons. That means the muon energy loss is dominated by

radiative processes at high energies.

2/ ~ ( / )c ce eE E m m

If the critical energy

was ~ 10 MeV for e,

then the critical

energy for muons will

be ~ 420 GeV. That

becomes important at

new colliders such as

the LHC

Tracker Material

For a complete understanding of the momentum scale and

resolution a detailed understanding of the tracker material

throughout the system is needed – use photon conversions

for high Z and nuclear interactions for low Z material.

radius

beam pipe, pixels, strips

50

Particle ID

51

2/3 2

2 2

2/3

1

~

~ ( )

/ ~ ( / ) / [5.1 ]

N

B

p

e

o

A

Z

X Z A A